Phase transitions of McKean–Vlasov processes in double-wells landscape

Abstract

We aim to establish results about a particular class of inhomogeneous processes, the so-called McKean-Vlasov diffusions. Such a diffusion corresponds to the hydrodynamical limit of an interacting particle system, the mean-field one. Existence and uniqueness of the invariant probability are classical results provided that the external force corresponds to the gradient of a convex potential. However, previous results, see , state that the non-convexity of this potential implies the nonuniqueness of the invariant probabilities under easily checked assumptions. Here, we prove that there exists phase transitions, that is under a critical value, there are exactly three invariant probabilities and over another critical value, there is exactly one. Under simple assumptions, these two critical values coincide and it is characterize by a simple implicit equation. We exhibit other cases in which phase transitions occur. Finally, we proceed numerical estimations of the critical values.

Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes

“…4], the McKean-Vlasov equation with a non-convex confining potential, can have several stationary solutions at low temperatures [13,44]. As a matter of fact, the number of stationary solutions depends on the number of metastable states (local minima) of the confining potential [45]. A complete rigorous analysis of phase transitions, both continuous and discontinuous, for the McKean-Vlasov dynamics in a box with periodic boundary conditions and for non-convex (i.e.…”

“…Based on earlier work [13,44], it is by now well understood that the number of invariant measures, i.e. the number of solutions to (7), is related to the number of metastable states (local minima) of the confining potential -see [45] and the references therein. For the Curie-Weiss (i.e.…”

Loss of Energy Concentration in Nonlinear Evolution Beam Equations

“…First, the rigorous multiscale analysis for the McKean-Vlasov equation in locally periodic potentials needs to be carried out. Perhaps more importantly, the rigorous construction of the bifurcation diagram in the presence of infinitely many local minima in the confining potential, thus extending the results presented in, e.g., Dawson (1983), Tamura (1984), Tugaut (2014), appears to be completely open. Furthermore, the study of the stability of stationary solutions to the McKean-Vlasov equation in the presence of a multiscale structure, as well as the analysis of the problem of convergence to equilibrium in this setting is an intriguing question.…”

“…2a, the oscillatory part of the potential introduces (infinitely many) additional local minima. Consequently, Tugaut (2014), the self-consistency equation R(m ; θ, β) = m has multiple solutions. Furthermore, as shown in Fig.…”

“…Based on earlier work (Dawson 1983;Tamura 1984), it is by now well understood that the number of invariant measures, i.e., the number of solutions to (1.7), is related to the number of metastable states (local minima) of the confining potential-see Tugaut (2014) and the references therein. For the Curie-Weiss (i.e., quadratic) interaction potential a one-parameter family of solutions to the stationary McKean-Vlasov equation (1.7) can be obtained: This one-parameter family of probability densities is subject, of course, to the constraint that it provides us with the correct formula for the first moment: R(m; θ, β).…”